After examining the spread of Islam, what areas were brought under its control and how?
The answer to this question depends a great deal on what sort of time period you are asking about. For example, Islam now “controls” Indonesia, but the faith did not come to dominate most of that country until almost 1,000 years after it was founded. For the purposes of this answer, I will focus on roughly the first 100 years after Muhammad founded Islam.
By 750 AD, Islam had come to control a very broad area. It had arisen, of course, in Arabia. From there, it had spread out to the east, the north, and the west. In the west, Islam had reached all the way to Spain. All of Iberia was dominated by Islam by about 720. On the way to Iberia, Islam had come to dominate all of North Africa as well. To the north, Islam controlled area all the way up to the region between the Black Sea and the Caspian Sea. Islam controlled all of the eastern shore of the Mediterranean. Its control reached all the way to what is now Afghanistan and Pakistan in the east.
Islam came to control this region largely through conquest. It is true that the ideas of Islam spread through trade and converted many people. However, Islam did not come to control these areas through voluntary conversion. Instead, it came to control them because it was militarily powerful and was able to conquer these regions.